3.95 \(\int \frac{\left (a+b x^4\right )^{3/4}}{c+d x^4} \, dx\)

Optimal. Leaf size=173 \[ \frac{b^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d}+\frac{b^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d}-\frac{(b c-a d)^{3/4} \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d}-\frac{(b c-a d)^{3/4} \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d} \]

[Out]

(b^(3/4)*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(2*d) - ((b*c - a*d)^(3/4)*ArcTa
n[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(2*c^(3/4)*d) + (b^(3/4)*A
rcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(2*d) - ((b*c - a*d)^(3/4)*ArcTanh[((b*c
- a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(2*c^(3/4)*d)

_______________________________________________________________________________________

Rubi [A]  time = 0.239638, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ \frac{b^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d}+\frac{b^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 d}-\frac{(b c-a d)^{3/4} \tan ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d}-\frac{(b c-a d)^{3/4} \tanh ^{-1}\left (\frac{x \sqrt [4]{b c-a d}}{\sqrt [4]{c} \sqrt [4]{a+b x^4}}\right )}{2 c^{3/4} d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(3/4)/(c + d*x^4),x]

[Out]

(b^(3/4)*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(2*d) - ((b*c - a*d)^(3/4)*ArcTa
n[((b*c - a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(2*c^(3/4)*d) + (b^(3/4)*A
rcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(2*d) - ((b*c - a*d)^(3/4)*ArcTanh[((b*c
- a*d)^(1/4)*x)/(c^(1/4)*(a + b*x^4)^(1/4))])/(2*c^(3/4)*d)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 37.9971, size = 148, normalized size = 0.86 \[ \frac{b^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2 d} + \frac{b^{\frac{3}{4}} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2 d} - \frac{\left (- a d + b c\right )^{\frac{3}{4}} \operatorname{atan}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{2 c^{\frac{3}{4}} d} - \frac{\left (- a d + b c\right )^{\frac{3}{4}} \operatorname{atanh}{\left (\frac{x \sqrt [4]{- a d + b c}}{\sqrt [4]{c} \sqrt [4]{a + b x^{4}}} \right )}}{2 c^{\frac{3}{4}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(3/4)/(d*x**4+c),x)

[Out]

b**(3/4)*atan(b**(1/4)*x/(a + b*x**4)**(1/4))/(2*d) + b**(3/4)*atanh(b**(1/4)*x/
(a + b*x**4)**(1/4))/(2*d) - (-a*d + b*c)**(3/4)*atan(x*(-a*d + b*c)**(1/4)/(c**
(1/4)*(a + b*x**4)**(1/4)))/(2*c**(3/4)*d) - (-a*d + b*c)**(3/4)*atanh(x*(-a*d +
b*c)**(1/4)/(c**(1/4)*(a + b*x**4)**(1/4)))/(2*c**(3/4)*d)

_______________________________________________________________________________________

Mathematica [C]  time = 0.25475, size = 161, normalized size = 0.93 \[ \frac{5 a c x \left (a+b x^4\right )^{3/4} F_1\left (\frac{1}{4};-\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )}{\left (c+d x^4\right ) \left (x^4 \left (3 b c F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )-4 a d F_1\left (\frac{5}{4};-\frac{3}{4},2;\frac{9}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )+5 a c F_1\left (\frac{1}{4};-\frac{3}{4},1;\frac{5}{4};-\frac{b x^4}{a},-\frac{d x^4}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x^4)^(3/4)/(c + d*x^4),x]

[Out]

(5*a*c*x*(a + b*x^4)^(3/4)*AppellF1[1/4, -3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c
)])/((c + d*x^4)*(5*a*c*AppellF1[1/4, -3/4, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)]
+ x^4*(-4*a*d*AppellF1[5/4, -3/4, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + 3*b*c*Ap
pellF1[5/4, 1/4, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))

_______________________________________________________________________________________

Maple [F]  time = 0.055, size = 0, normalized size = 0. \[ \int{\frac{1}{d{x}^{4}+c} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(3/4)/(d*x^4+c),x)

[Out]

int((b*x^4+a)^(3/4)/(d*x^4+c),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{d x^{4} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/(d*x^4 + c),x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(3/4)/(d*x^4 + c), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.29378, size = 1006, normalized size = 5.82 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/(d*x^4 + c),x, algorithm="fricas")

[Out]

-((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)/(c^3*d^4))^(1/4)*arctan(c^
2*d^3*x*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)/(c^3*d^4))^(3/4)/(x
*sqrt(((b^3*c^4*d^2 - 3*a*b^2*c^3*d^3 + 3*a^2*b*c^2*d^4 - a^3*c*d^5)*x^2*sqrt((b
^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)/(c^3*d^4)) + (b^4*c^4 - 4*a*b^
3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(b*x^4 + a))/x^2) + (
b*x^4 + a)^(1/4)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2))) + (b^3/d^4)^(1/4)*arctan(d^3*
x*(b^3/d^4)^(3/4)/((b*x^4 + a)^(1/4)*b^2 + x*sqrt((b^3*d^2*x^2*sqrt(b^3/d^4) + s
qrt(b*x^4 + a)*b^4)/x^2))) - 1/4*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3
*d^3)/(c^3*d^4))^(1/4)*log((c^2*d^3*x*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2
- a^3*d^3)/(c^3*d^4))^(3/4) + (b*x^4 + a)^(1/4)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2))
/x) + 1/4*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)/(c^3*d^4))^(1/4)*
log(-(c^2*d^3*x*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)/(c^3*d^4))^
(3/4) - (b*x^4 + a)^(1/4)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2))/x) + 1/4*(b^3/d^4)^(1
/4)*log((d^3*x*(b^3/d^4)^(3/4) + (b*x^4 + a)^(1/4)*b^2)/x) - 1/4*(b^3/d^4)^(1/4)
*log(-(d^3*x*(b^3/d^4)^(3/4) - (b*x^4 + a)^(1/4)*b^2)/x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{c + d x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(3/4)/(d*x**4+c),x)

[Out]

Integral((a + b*x**4)**(3/4)/(c + d*x**4), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{d x^{4} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/(d*x^4 + c),x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)/(d*x^4 + c), x)